Theorem imaeq2 5018

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4954	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4907	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4688	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4688	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1373   ran crn 4676   |` cres 4677   "cima 4678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by:  imaeq2i  5020  imaeq2d  5022  fimadmfo  5507  ssimaex  5640  ssimaexg  5641  isoselem  5889  f1opw2  6152  fopwdom  6933  ssenen  6948  fiintim  7028  fidcenumlemrk  7056  fidcenumlemr  7057  sbthlem2  7060  isbth  7069  ennnfonelemp1  12777  ennnfonelemnn0  12793  ctinfomlemom  12798  ctinfom  12799  tgcn  14680  iscnp4  14690  cnpnei  14691  cnima  14692  cnconst2  14705  cnrest2  14708  cnptoprest  14711  txcnp  14743  txcnmpt  14745  metcnp3  14983